Galileo's Paradox
in [Math]
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From: Virgil on 30 Nov 2006 16:49 In article <456f3385(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456e46b7(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Any uncountable > >> set with element indexes expressed as digital numbers will require > >> infinitely long indexes for most elements, such as is the case for the > >> reals in any nonzero interval. > > > > There is nothing in being a set, including being a set of reals, that > > requires its members to be indexed at all. > > To establish an explicit bijection between infinite sets does require an > ordering on the sets, at least in general. It requires a function between the sets, but neither set need be ordered. For example, let P = R^2 be the Cartesian plane, then for any reals a and b with a^2 + b^2 > 0, (x,y) |--> (a*x + b*y, -b*x + a*y) bijects P to itself, but P is not an ordered set.
From: Virgil on 30 Nov 2006 16:51 In article <456f34bc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456e475e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Given a > >> set density, value range determines count. > > > > Compare the "set densities" of the set of naturals, the set of > > rationals, the set of algebraics, the set of transcendentals, the set of > > constructibles, and the set of reals. > > Rather difficultt o formulate relations between those in standard > theory. In the name of IST, I'll avoid any criteria including the notion > of "standard" and state the following. The size of the set of > hypernaturals is the square root of the size of the set of hyperreals. > The set of hyperrationals corresponds to the square of the set of > hypernaturals, minus all those pairs that are redundant, such as 2/4 or > 6/18. That number of the hyperreals are the hyperirrationals. I am not > sure how to relatively quantify transcendentals, constrictibles, or > algebraics. Those are probably considered all "countable" by you, which > doesn't say much about their relative sizes. When challenged to support his fool theories, TO resorts to nonsense.
From: stephen on 30 Nov 2006 17:11 Six wrote: > On Thu, 30 Nov 2006 17:44:46 +0000 (UTC), stephen(a)nomail.com wrote: <snip> >>Six wrote: >> >>> I suspect though that there may be no proof of the matter either >>> way, as has been hinted at in other parts of this thread. It may come down >>> to this: that someone who wants to take a different, but still very >>> reasonable (maybe more reasonable) appoach to this paradox, would need to >>> demonstrate that some interesting and viable mathematics can result from >>> it. This would certainly involve having precise defintions and so forth; it >>> is just that they would be more or less different ones. I certainly do not >>> have the wherewithall to even begin such a task. But it's interesting to >>> speculate. And it's good to keep an open mind. >> >>You are more than welcome to come up with a definition of 'how many' >>that you like. If other people like it, they may use it. It is not >>going to change the fact that the naturals and the squares have the >>same cardinality, nor the fact that the squares are a proper subset >>of the naturals. >> > That is exactly what it is going to change. How can that possibly change? There exists a bijection between the naturals and the squares, therefore the two sets have the same cardinality. Every square is a natural, and some naturals are not squares, therefore the squares are a proper subset of the naturals. Those two simple facts cannot change no matter what definition of 'how many' you come up with. > Your insistence that I define my terms in advance is back to front. > There is a primitive intuition that naturals exceed squares and that they > are equinumerous. There is no simple confusion here. Though undoubtedly > there is something to be learned. If you are not going to define your terms, then what you say is meaningless. You say 'how many' but you refuse to say what you mean by that. How is anyone supposed to understand you? > It seems to me that your understanding of the unclarity about how > many is a phantom product, a reflection of the set theory you accept so > uncritically. You want to pretend the ambiguity about how many pre-exists, > whereas in fact it is a creation of the mathematics. Conventional set > theory has no unique. proprietary rights over this paradox. and it seemed > to me that, whatever else it does and no doubt does very well, it deals > with this paradox rather poorly. > Thanks, Six Letters I am not claiming any unique proprietary rights over this paradox. I am simply claiming that it is only a paradox because you are relying on vague notions of 'how many' and 'size'. There is absolutely nothing paradoxical about the statement Every element of S is an element of N, but N contains elements not in S, and There exists a bijection between S and N. That is all we are saying when we say that S is a proper subset of N, and S has the same cardinality as N. That is all set theory says on the subject. The only problem is that you have some vague notion of 'size' or 'how many' that you are applying. But you apparently refuse to even look at your notion of 'size' or 'how many', and instead would rather complain about others being close minded. Stephen
From: Lester Zick on 30 Nov 2006 17:40 On Thu, 30 Nov 2006 17:44:46 +0000 (UTC), stephen(a)nomail.com wrote: >Six wrote: [. . .] >What do you mean by a 'good definition'? What makes something a good >definition, as opposed to a bad definition. It helps if the definition is true. ~v~~
From: Michael Press on 30 Nov 2006 18:21
In article <1164888031.386857.96190(a)j72g2000cwa.googlegroups.com>, "Tonico" <Tonicopm(a)yahoo.com> wrote: > Ps Have you, and anyone else, noted how all the anticantorian cranks > are NEVER mathematicians? But Internet welcomes all, and google's > sci.math is an uncensored group, so anyone can offer his piece to > all...and you know what? I think this is just fine. I'm convinced that > also from the most stupid, dense and even annoying crank/troll we all > can learn. I too conduct my affairs in accord with this conviction. -- Michael Press |